the order of an element, has to divide the order (size) of the entire group of units.

for 11, U(11) has 10 elements, so the possible orders of elements of U(11) (which are all the integers mod 11 save 0, because 11 is prime), are 1,2,5 and 10.

only 1 will have order 1. 2^2 = 4 ≠ 1, so 2 doesn't have order 2. 2^5 = -1 (or 10, doesn't matter), so 2^5 does not have order 5. that just leaves 10, and 2^10 = 1, as expected

(all these values are mod 11, of course).

now U(17) has 16 elements. possible orders are 1,2,4,8 or 16.

while it is true that 2^16 = 1, that does not mean 2 has order 16. why? because 2^8 = 1, and 8 < 16, so 2 has order 8.

not every element of U(17) will have order 16. for example 2^4 = 16 (mod 17). as you can easily see, 16 only has order 2.

the upshot of this, is that 2 does not generate the entire set of units via powers, the powers of 2 are:

{2,4,8,16,15,13,9,1}, which is only half of U(17) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}.

November 9th 2011, 11:44 PM

Opalg

Re: Primitive root question

How do you find a primitive root of 17? Well, you have already found that 2 has order 8. So if you can find an element x such that then x will have order 2*8=16 in U(17) and will therefore be a primitive root.